Can I prevent buyers from sharing or having extra copies of my file?

After someone has downloaded a file successfully just once, it is impossible to prevent them from sharing that file or making additional copies of that file from their own computer, without even having to "download another copy".

What about restricting download Attempts?

Since restricting download Attempts has no effect on the downloader's ability to make additional copies on their own after their first successful download, the only purpose of setting download Attempts is to define how soon the link expires and becomes useless to everyone;

It is also impossible to determine if any of the Attempts made on a download link resulted in a completed download file, nor if any completed download resulted in a valid, usable, openable file -- a wide variety of uncontrollable technical glitches can result in any download attempt failing or resulting in a corrupt/damaged or othewise useless file;

Some antivirus/malware scanners will pre-fetch links to look for computer threats, so that would count as at least one Attempt before a buyer even sees their link, and some "download accelerator" toolbars/add-ons can burn up multiple Attempts at once which could instantly expire a buyer's link as soon as they try downloading -- see further details here;

The more restrictive your download expiration settings, the more time you will spend dealing with honest, frustrated customers who simply needed more Attempts to complete even one download successfully, and the greater your rate of refunds/reversals will be, so you may wind up giving back more money than you'd ever lose due to file sharing;

Many buyers who are not given enough chances to complete a download successfully will not bother to contact you and simply file an immediate dispute with the payment processor or their card issuer, who generally side with the buyer in dispute resolutions for digital-product transactions, and excessive disputes could get your payment account restricted or suspended.

Other considerations

Unlike tangible items, where each unit has a specific production/wholesale cost that is lost for each unit stolen, your "sunk cost" for producing digital files is the same whether that file exists as 10 copies or 1000 copies, regardless of how many copies were purchased vs. shared, so file sharing doesn't really "cost" you anything but potential lost sales;

Few buyers are patient and tech-savvy enough to figure out how to use file sharing services in hopes of finding a file they may want, and few who would accept or seek out shared copies for free would ever willingly pay for them, so most of those cases cannot really be considered "lost" sales if they'd never buy to begin with;

File sharing mostly only affects sales of corporate blockbuster/bestseller titles in music, movies, games, and software, which more people would otherwise be willing to pay for if they couldn't find it for free, but most other, less popular titles are rarely even posted to file sharing sites at all;

Treating every legitimate buyer like a potential criminal does nothing to prevent file sharing, but it does increase your customer service workload and degrade your customer satisfaction and reputation;

If time = money, you could wind up wasting a lot of both dealing with customer complaints, refunds/reversals, reactivating/reissuing links, and defending your reputation.

In a nutshell, make your product easier to buy legitimately than to find on file-sharing services -- if you price your product fairly and make it easy to obtain and use, more people will be willing to buy it for instant gratification, rather than going through the hassle of scouring file-sharing services in hopes of finding a free copy. For example, standup comic Louis CK recently offered a download of his full-length comedy special for only $5; within less than two weeks, he'd already made $1 million, despite having no inherent restrictions to impede sharing his downloaded file.

Is there some way I can control who can open an eBook, even if it gets shared?

If you are selling eBooks, there are solutions to control the ability to open a file, so any copies would be useless without the access code/key. We also have a PDF Stamping feature that presents a strong deterrent against file sharing by putting the buyer's name, email and unique Transaction ID on every page -- please see Selling eBooks with E-junkie for more details.